Integrand size = 25, antiderivative size = 107 \[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\frac {\operatorname {AppellF1}\left (\frac {1-m}{2},1,-p,\frac {3-m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \cot (e+f x))^m \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)} \]
AppellF1(-1/2*m+1/2,1,-p,3/2-1/2*m,-tan(f*x+e)^2,-b*tan(f*x+e)^2/a)*(d*cot (f*x+e))^m*tan(f*x+e)*(a+b*tan(f*x+e)^2)^p/f/(1-m)/((1+b*tan(f*x+e)^2/a)^p )
Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(107)=214\).
Time = 2.10 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.48 \[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=-\frac {a (-3+m) \operatorname {AppellF1}\left (\frac {1-m}{2},-p,1,\frac {3-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x) \cot (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p}{f (-1+m) \left (-2 b p \operatorname {AppellF1}\left (\frac {3-m}{2},1-p,1,\frac {5-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 a \operatorname {AppellF1}\left (\frac {3-m}{2},-p,2,\frac {5-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a (-3+m) \operatorname {AppellF1}\left (\frac {1-m}{2},-p,1,\frac {3-m}{2},-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cot ^2(e+f x)\right )} \]
-((a*(-3 + m)*AppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a ), -Tan[e + f*x]^2]*Cos[e + f*x]^2*Cot[e + f*x]*(d*Cot[e + f*x])^m*(a + b* Tan[e + f*x]^2)^p)/(f*(-1 + m)*(-2*b*p*AppellF1[(3 - m)/2, 1 - p, 1, (5 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + 2*a*AppellF1[(3 - m)/2, -p, 2, (5 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x]^2] + a*(-3 + m)*A ppellF1[(1 - m)/2, -p, 1, (3 - m)/2, -((b*Tan[e + f*x]^2)/a), -Tan[e + f*x ]^2]*Cot[e + f*x]^2)))
Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3042, 4157, 3042, 4153, 395, 394}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (d \cot (e+f x))^m \left (a+b \tan (e+f x)^2\right )^pdx\) |
\(\Big \downarrow \) 4157 |
\(\displaystyle \left (\frac {\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \int \left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (b \tan ^2(e+f x)+a\right )^pdx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \left (\frac {\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \int \left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (b \tan (e+f x)^2+a\right )^pdx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \frac {\left (\frac {\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \int \frac {\left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (b \tan ^2(e+f x)+a\right )^p}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 395 |
\(\displaystyle \frac {\left (\frac {\tan (e+f x)}{d}\right )^m (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \int \frac {\left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^p}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 394 |
\(\displaystyle \frac {\tan (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1-m}{2},1,-p,\frac {3-m}{2},-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f (1-m)}\) |
(AppellF1[(1 - m)/2, 1, -p, (3 - m)/2, -Tan[e + f*x]^2, -((b*Tan[e + f*x]^ 2)/a)]*(d*Cot[e + f*x])^m*Tan[e + f*x]*(a + b*Tan[e + f*x]^2)^p)/(f*(1 - m )*(1 + (b*Tan[e + f*x]^2)/a)^p)
3.5.22.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 , -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ FracPart[p]) Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ [{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && !(IntegerQ[p] || GtQ[a, 0])
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
Int[(cot[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + ( f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[(d*Cot[e + f*x])^FracPart[m]*(Tan [e + f*x]/d)^FracPart[m] Int[(a + b*(c*Tan[e + f*x])^n)^p/(Tan[e + f*x]/d )^m, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[m]
\[\int \left (d \cot \left (f x +e \right )\right )^{m} \left (a +b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
\[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
Timed out. \[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\text {Timed out} \]
\[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
\[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int { {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
Timed out. \[ \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx=\int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \]